Why it happens: Since the sum of the first n odd numbers is n2, going from 352 to 362 means adding just one more term of that series. The same step in algebra: (n + 1)2 = n2 + 2n + 1, and here 2 × 35 + 1 = 71.
List the odd numbers with their positions and the rule appears at once.
Position n
1
2
3
4
5
6
Odd number
1
3
5
7
9
11
Each odd number is one less than twice its position: 2 × 6 – 1 = 11. So
36th odd number = 2 × 36 – 1 = 71
Why it happens: The even numbers in order are 2, 4, 6, …, so the n-th even number is 2n. Every odd number sits exactly one step below an even number, which gives 2n – 1.
Q3.
What is the nth odd number?
Answer
The n-th odd number is 2n – 1.
n = 1 → 2(1) – 1 = 1 n = 5 → 2(5) – 1 = 9 n = 50 → 2(50) – 1 = 99
Why it happens: Any odd number leaves remainder 1 on division by 2, so it can be written as 2k + 1 for some whole number k ≥ 0. Counting from k = 0 gives 1, 3, 5, …, and the term in position n has k = n – 1, so the value is 2(n – 1) + 1 = 2n – 1.
Tip: This formula is what makes the odd-number test practical. Adding 1 + 3 + 5 + … + (2n – 1) gives n2, so the position at which you land on 0 while subtracting is the square root.